Yh I should have been more clear on meaning that it is a monopoly of the state, which includes the Fed and the high level private corporate interests like private bansks, monopolies and cartels. What you've highlighted is the corporatist nature of the modern capitalist state.

This highlights the ambiguous nature of private financial institutions, which often seem to blend into state institutions, as you note when you highlight how the Fed's shareholders are private banks (but, ultimately, it is the central bank which creates all the money which those firms then end up with and circularly invest or save with the central bank) and that those banks appoint board members. Its a revolving-door world because they are all part of the capitalist state. I'm not speaking just about the Fed, which is one institution within the state, but which plays a key role in the creation and managing of money. Money itself is always a monopoly of the state. The state, in the marxist sense, is not simply the government or the narrow set of institutions that liberals refer to as the 'state'. Its an organic, necessary type of structure or institution which is an expression of capitalist class power. The largest corporations cannot be fully separated from the state, imo.

The capitalist state needs money, and its monopoly of control over it (dramatic examples in the case of quantitative easing), in order to reproduce capitalist social relations, i.e. generalized commodity production, because that is the purpose of the capitalist state and money is a necessary expression or form of capital under capitalism. The fetishitic belief in the value of money as the universal expression of value (which itself makes money socially, objectively real - comrades are wrong in a really fundamental way when they say that money isn't real) it key to the functioning of capital and for the type of social organization and planning (which, in financial capitalism, develops more and more closely towards a form of centralized planning centralized in the financial sector, aided by the state institutions). The capitalist needs money for universalized price systems so that they can get measures of value (socially necessary labour time) and thus profitability (i.e. to guide them superficially to maximising the rate of profit : r = s/ (c + v) ). They need a common measure for that which is ultimately common to every commodity: abstract labour power / socially necessary labour time. The worker ultimately needs it because they have only their labour power to sell, which cannot easily be bartered, meaning they need a universal means of trade for their own reproduction. The demand for money is thus endogeneously universalized within capitalism. It is inextricably linked to capitalist social power. The demand for it, and the belief in its value, ensures the objective economic domination of the capitalist class over us. If necessary, the state ,which, structurally, exists necessarily to support conditions of capitalist social reproduction, will act against the interests of individuals capitalists and firms if they take it to be in the interest of capitalism as a whole, for instance letting banks go bust while trying to avoid systemic risk leading to the collapse of the functioning of all of the financial sector and monetary system. In this way it doesn't act like other, private, capitalist social organizations.

Its because they need it and control over it that they will never accept ancap, libertarian pipe-dreams of allowing anyone from your mother to your priest to your dog to set up a competing currency, leaving aside the reasons why that is a ridiculous idea anyway.

A contradiction is that capitalism needs a state which maintains monopoly over this public asset - money - even though it is inevitably determined, used in the service of, and ultimately controlled, especially since the neoliberal reaction, by private capitalist interests, while it is also clearly socially rational to bring any such institution under public control. Of course that would be inconsistent with capitalism and (hopefully) imply a socialist economy under a socialist state directed by a democratic proletarian dictatorship.

You highlight the importance of this growth of corporatism which is so natural to the finance-monopoly imperialist stage of capitalism, especially in its rejuvenation in the reaction of neoliberalism. This is setting the stage for the highest stage of imperialism - fascism The explicit power of corporations, especially financial and multinational monopolies, in whose interest the state and thus the monetary system is wielded, further intensifies commodifcation, exploitation, alienation, postmodern culture, cynicism, and sets the state for the reaction against international finance capitalism that elements of the capitalist state will then exploit and use to, paradoxically, support itself by more fully actualizing fascism.

I think its worth noting though that there are real tensions between different parts of the ruling classes regarding how to use the monetary system. Many national capitalists and business owners who have to engage in real production and are not simply investing in financial assets have not been happy with the monetary tightening, because it limits their access to credit and increases their interest rates, as well as slowing down an economy they depend upon for demand. It has been more in the interest of the financier class. The coup of Truss within the Tory part, installing a financial sector puppet like Sunak, was done for similar reasons. This simply reflects a contradiction between national and international or financial capitalists, although I think we sometimes exaggerate the extent to which this is reflected in differences in party-support. Obviously again important for understanding fascism because the national bourgeoisie and petit-bougies whose interests contradict those of the international finance-monopolists in some respects themselves play key roles historically in supporting and benefitting from fascism.

Edit: TLDR - the exclusive normative universality of money as a measure of value (which is socially objective, it really structures society) is enforced and ensured by the exclusive universality of the state and its monopolies on violence and money itself.